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OCR for page 455
Some Numenca] Computations about Free Surface Boundary Layer
and Surface Tension Effects on Nonlinear Waves
E. Campana
University "La Sapienza"
Roma, Italy
F. Lalli and U. Bulgarelli
I§tituto Nazionale per Studi ed Espenenze di Architettura Navale
Roma, Italy
Abstract____
This paper is a first approach to
the development of a full nonlinear
numerical method implementing the
exact boundary conditions, taking into
account viscosity and surface tension
effects at the free surface. At this
step, the following results have been
obtained, maintaining the simplicity of
the collocation method proposed by
Dawson t53:
1) effective fully nonlinear
numerical procedure, taking into
account the effects of viscosity at the
free surface on potential flow;
2) simple preliminary results for the
linear gravitycapillarity problem.
All the obtained numerical results
have been tested with analytical
solutions or experimental results.
List of symbols
0xy = frame of reference fixed with
the body : Ox is oriented in
the opposit direction of the
body velocity V, Oy is
vertical upwards
= body surface
L = characteristic body length
(chord)
depth of the body center
free surface
unit vector normal to and S
Cartesian equation of the
free surface
curvature of the free surface
curvilinear abscissa defined
along S
free surface boundary layer
thickness
body velocity vector
h
S
n =
tax) =
K =
1 =
=
=
~(U,O) =
T =
k 

v =
=
g =
P =
Rw =
=
=
of =
r =
r* =
P =
~ =
q* =
455
= fluid velocity vector
= vorticity vector
= density
surface tension
wave number
group velocity
kinematic viscosity
acceleration of gravity
pressure
wave resistance
velocity potential
perturbation potential
simple layer density
p ql
IP  q *1
field point
source point
image source point
1. Introduction
The Rankine source method
succesfully adopted in recent
the numerical solution of
free surface flows.
In the usual approach the free
surface boundary conditions are
linearized and applied on the
undisturbed free surface. Then, using
the simple layer potential function
the boundary value problem is r"
to an integral equation which
solved numerirn11v
has been
years for
potential
can be
_ _ _ _ i, .
With this approach, in 1977 Dawson
t53 proposed a simple and effective
procedure for computing the wave
resistance of ships in the steadystate
case. Such procedure was based on the
idea of writing the above mentioned
linearized boundary conditions along
the zero Froude number flow streamlines
on the undisturbed free surface.
In the wake of Dawson's paper
several studies have then appeared in
literature; some Authors deal with the
linear problem, discussing about the
consistency of different kinds of
OCR for page 455
linearizations t9,13], the numerical
implementation of the radiation
condition [10,12,14] or the numerical
stability [123.
On the other hand some Authors,
maintaining more or less the simplicity
of the collocation method, and the use
of an upstream finite differences
scheme t5] for the radiation condition,
proposed some kinds of iterative
procedures to solve the nonlinear
problem [6,11,153.
The present paper is a first
approach to our future aim to develope
a full nonlinear method, implementing
the exact boundary conditions taking
into account viscosity and surface
tension effects at the free surface.
This combined gravitycapillarity
problem is interesting because can give
some more informations on nonlinear
waves behavior, with respect to the
classical model which neglects surface
tension. As a matter of fact, Longuet
Higgins r 2] proposed a model for the
nonlinear transfer of energy from steep
gravity waves to capillary ripples,
which "ride" on the forward faces of
the long waves. In this case the
surface tension can be locally
important and play a significant role
in the generation of waves by wind.
Furthermore capillary waves, taking
energy from gravity waves and loosing
it by viscosity,tend to delay the onset
of breaking.
On the other hand Marco and
Ikehata, in an experimental work [8],
pointed out the importance of surface
tension on the shape of the wave
pattern around the model, in particular
at the forebody; how wave resistance
coefficient depends also on the Weber
number is furthermore shown in t83.
Among the free surface phenomena,
the presence of a thin boundary layer
can also be considered. All real fluid
motions are of course rotational; even
in nearly irrotational flows the
relatively small amount of vorticity
present in thin layers, can be crucial
in determining the main flow
characteristics, as it's well known.
The condition of vanishing of the
tangential stress at a perturbed free
surface cannot be satisfied by
irrotational motion. Consequently
regard the incompressible flow as
combination of a thin rotational free
surface layer and a potential flow
region. The exact amount of the
vorticity jump generated at the free
surface can be written as a function of
its curvature and of the potential
velocity t4~.
So we obta~
boundary cordite
effect at S; o
an
we
a
in a nonlinear dynamic
ion including viscosity
f course in the limiting
456
case of Re ~ on , the
condition is obtained.
Numerical results have been
computed for the full nonlinear
formulation in the presence of a free
surface boundary layer, while simple
linear results are presented for the
gravitycapillarity inviscid problem.
In order to validate the
capability of the proposed model,
numerical simulations of some typical
test problems are described and
discussed.
usual Bernouilli
_. Mathematical formulations____________ ____________
We consider the following
mathematical formulation for a 2D
steady state potential flow, due to the
motion of a submerged nonlifting body
in a fluid of infinite depth. The
extension to the 3D case can be easily
obtained. The velocity V=(U,O) of the
body is directed in the negative x
direction, the y axis is positive
upwards and the undisturbed free
surface level is given by y=O.
Consequently the fluid domain ~ is
bounded on the upper part by a free
boundary S. and it is unbounded in the
other directions. We assume that the
fluid velocity, q=(u,v), can be written
as:
(1) q—V,
where
(1 ') ~ = ~
In (1') the term LTx is the undisturbed
flow potential and the term Stakes
into account the interaction between
the free surface and the body. Let be
y= ~ (x) the Cartesian equation of the
free boundary S. and l a curvilinear
abscissa defined along S.
The potential ~ (x,y) satisfies
Laplace's equation inside ~ `~ , where
= ~ V ~~ C ~2 is the body
(2) V {(x,'~sO ~ (x'~6 t`6 n T. I}
The following boundary conditions,
which include the effects of surface
tension T , must be associated to
equation (2) :
( ) lt orb 49 on S
OCR for page 455
~ (em :~)+ T ~ on S
2~se .~)
(5) ~ ~ O

(6) km  V i bounded
~x'_oo
on ~o3
Here K(l) is the curvature of S.
considered negative when the centre of
curvature lies on the fluid side of the
free surface:
en
(7) K=.~ Ax
(4~ ,2 )3~
x
From now on, in
will present, an
conditi _
of ~ whenever the d
to have finite depth.
We are going to c
kinds of approximation
previously described,
boundary conditions
obtained from.(3) and
for computational
precisely four models
every model we
extra boundary
on must be imposed at the botton
omain is supposed
Therefore, in this case, the linear
model describing the motion of a
submerged body is constituted by (2),
(5), (6')~9) and (8) or (3'). It is
worth to notice that, for this
formulation, theorems for solution
existence and uniqueness have been
given in r7]
2.2 The linear formulation with T/O
Without neglecting surface
tension, conditions (9) and (8) can be
written as:
(10) ~—YXxi t,= ~ ', _ T
rr
(11) A= _ u ~ 4,
1 ~ x pit L~
Hence, in this case s the linear model
is given by (2), (5), (6), (10) and
(11) or (3').
Onsider different
s for the problem
in which the
given on S. In the nonlinear case, with T=0,
(4), are suitable condition (9) becomes:
purposes. More
are delineated.
z.3 The nonlinear formulation with T=0
z.1 The linear formulation with T=0
(12) ~ ~ ~ §~' = 0 on S
while the nonlinear Bernouilli equation
give:
Assuming T=0 in (4) and neglecting
the nonlinear terms both in (3) and
in(4), we get the linearized free  ~ / UP &2 ~
surface conditions: (13) ~ ~ ~ ~ ~ ~4J on S
(3~) U ~ =7, on y=0
(8, ~ 5 _ U ~ on y=0
and the well known NeumannKelvi n
condition:
(9) U y +~=0
~ xx g
where, for the car __.
we have (see 2.6):
(6') km Ivy ~ IT
x ~ — oo
di t i on At infinity,
Therefore the nonlinear model,
neglecting surface tension, is given
by equations (2), (5), (6), (12) and
(13) or (3)
2.4 The nonlinear formulation with TWO
Assuming TWO, the unified free
surface condition is obtained by
eliminating ~ between (3) and (4~:
(14) ~ ~ ~ ~ ~ = T t~
So the nonlinear model with surface
tension is given by equations (2), (4),
is), (6), (14). It is worth to observe
that conditions (9) , (10) , (12) can
be easily deduced from (14~.
In the sequel we confine ourselves
to consider the models given in 2.3
on S
457
OCR for page 455
and 2.2.
2.5 The effect of viscosity at the
free surface
Let us consider the model
described in 2.3, in which we will
introduce, in the hypothesis of large
Reynolds numbers, the modifying effect
of viscosity.
The introduction of a boundary
layer at the free surface can be
considered as a first step of a zonal
splitting of the global problem into
several interacting ones:
 the irrotational subdomain;
 the boundary layer at the free
surface;
 the boundary layer about the body km 1q = U
 the wake; 1 a= Ho
Furthermore the new
dynamic
boundary condition improves some
convergence properties of the numerical
procedure t153.
To deduce the above mentioned
condition we start from the
consideration that the irrotational
motion does not fulfil the condition of
zero tangential stresses at the free
surface (if the free boundary has a non
zero curvature). Consequently, we
introduce into the model given in 2.3
a partition of the flow field into an
inviscid region, free from vorticity,
and a thin viscous layer at the free
boundary. Across the second region a
vorticity jump, connected with the
curvature of S. is considered, as well
as the consequent velocity jump, due to
the viscous component inside the
boundary layer. In fact, although a
rigid boundary is the commonest source
of vorticit,~, in the case of a free
boundary the vanishing of the
tangential stresses generates vorticit~
and consequently a viscous boundary
layer. Across this layer the above
mentioned finite jumps of the velocity
and of the vorticity are generated.
To include the effects of the
vorticity at the free surface, we
start from the steady NavierStokes
equation:
(~5) [(at V)] = ~ V ~ ~ ~ ~q !21) ~< ~ ^¢
where p = p + 9~5 , p is the
dynamic pressure, p Is the atmospheric
pressure and ~ is the fluid density.
By using the following identities:
(16') (q V)9 = V(2q 9) 9 _
(16'') ~q = ~ VX
and, for y= ~q, (x):
(171 V (42 q q §~) 9 ~ ~
Since the integral on the left hand
side of (17) is path independent, we
can integrate both sides of (17) along
S. which in the steady flow is a
streamline. Requiring that:
we obtain:
~ , ~ ~ 9t 2 9 9 A ~vxz
since qx4~de =0 .
The kernel of the curvilinear
integral can be trasformed as follows:
( 1 9 ) (Van; )~t =  — d!
_ ~—
where n is the unit vector normal to S.
oriented outwards.
By substituting (19) into (18), we can
Basils ~et:
(20) 7~= Us_ ~ q q + ~ ~ ~; ~e'
2, &, ~ ~ ~ _ ~ ~
The normal derivative of ~ , if
the viscous layer is sufficiently thin,
can be expressed as the ratio between
the vorticity jump across the boundary
layer and its thickness`:
Con we shall use Helmholtz
cle.on~position and a boundary layer
approximation to evaluate the above
mentioned vorticit: Inch velocity jumps,
ill terms of the velocity potential
`>raciient and of the geometrical
curvature of the flee surface. The
458
OCR for page 455
expression of ^t can be readly
evaluated [43 as:
(21') at = 2 ~ vI
Moreover an extimation
oscillatory boundary layer
can be given r4~ by:
(22) ~ = (U!V)
Furthermore we observe
velocity distribution in
domain can be decomposed as:
~ vT qua
where q is required to satisfy:
or
V'9`r=0 Vx9~~t
of the
thickness
that the
the fluid
In particular the velocity at the free
boundary can be thought as the sum
of [V\' and the jump ^9~ of the
rotational component across the
boundary layer. This jump can be
estimated r 43:
(23) ~15~1 ~ ^:

hence, we have:
(23') Iql = IVII+AI6Vrl KIWI +
+` a:  IvII +~s K 1~74 ~
B>' introducing (21),(_2),(23~) in (20),
neglecting terms of order ~ we get:
(24) 9(~)= U _ ~4 ]' (~+ IS)+
+ ~ ~ 5 K(4) {~.~ And hence:
The unified free surface boundary
condition is obtained from (3), `24):
(25) j~4~ (~+4&K)~: 4~ ~
~ ~ me (SKI (' _ ~ K)  o
Ve observe that the usual in~iscid
nonlinear dynamical condition can lye
obtained from (25) in the limiting case
of ~ ~ O (i.e. Re. oo).
2.6 An integrodifferential formulation
for the gravitycapillarity problem
Let's consider now the linear
problem with surface tension described
in 2.2. We are going to develope some
mathematical calculations in order to
obtain an integrodifferential formula_
tion, in which the third order derive_
tive present in (10), that create some
difficulties for numerical implements_
tion, does not appear. Moreover, we
intend to explore the behavior of the
solution.
Solving (11) with respect to
it follows:
(26) ~xx~ 95 ~ = iT ~x= ~
that is:
(27) qtx)= ~ X LAI ~ ~ Je OFFS) d'] ~
o
~ 6dX[A2 ~ J~'~)di]
o
where An and A are arbitrary
constants, and ~ = IT
Since ~ is bounded for l~1. no, we
find:
^1 ~ i ~ ,rC')d;
A2 =  ~ J ~ Id) Hi
Cam
CO
(28) 4(X) _ ~ [I JO )~)di +
~ ~xJ ~, rt;,4~]
At this point, three different
formulations involving respectivel>
~ , by and ~~, will be obtained
Substituting the Expression of P (if)
and integrating by parts we ~et:
459
OCR for page 455
(29) ~q(x)= I—_:Q We STY 'by, _
x
Q J e ~ 1J~ d) ~
so
From (29), deriving with respect to x
and applying the kinematical condition
( 3 ' ) :
( 3 0 ) ~ ~ _ ~ ~ ~ 5~ [Q~XJ4,~l Ad
x
JO
The formulation involving y
fol lows directly from (28), de Diving
with respect to x and `'sing again
condi t ion ( 3 ' ):

~ _ _ ~ ~ dxJ d) g'
e J e ~l~ ~~]
_ ~
Finally, to
wi th y
(28): ),
get the last express ~ on
we integrate by parts
(32) ~(x)~~ g~`{x 2g,[ 1 Bit,'
~ ~ r ~ ~' ~
and the unified condition is obtained
as above:
~ 3 3 ) y _ _ p IJ [~d ~ J e ~' ~ 3)
AX Ix ^' ~
The relations ~ 3~)), ~ 31 ) or
can be used instead of ~ 10 I, to
the calculati. in of the third
~eriNiRti\'e yxxy G! yyyy in
u me r i c a l i m p l a m e n t a t i 0 !~ .
We notice that: f'~n~ f l) ) w
(I?) when
In o~
`~f the So'
f ~ee wave
f~ 1 1 '1; t, y
a I i ~ b l e s
cons tent s:
f ~ ~ )
i~Voi d
t? r ~ e r
the
... ~ obtai n
1 ma? () , t ha t i ~ ~ ~ on .
den to stlldv some properties
ution' we co~si<1e~ the ~ Ample
problem, given by ( 2 i, ( 10 ),
means of the epa~at ion off
He can ~;~ite, up a~bit~~
t 34 ) (p  Me { ~ ~ ~
where k is the separation constant.
Substituting (34) in (10) we get the
dispersion relation:
(35) ~ _ MU ~ ~ 9~=0
r T
u i t h: 4~' = ]°]J (A _ ~ )
C ~ (A ~ ~— ~ )
Hence, if
respectively:
be= ~
tic ~ (pure capillary waves )
Since we consider the steady case,
the phase velocity is zero, while the
group velocity is given by:
g or T ~ O we obtain
(pure gravity waves)
(36) ~ = ( _ _ ~ ) U
Uk
which can be written'
(37) ~ = ~ (I ~ )
4kT
 Us
The positive sign must be
gravity waves, and the negat
capillary waves: it is
recognize that the
positive for
460
chosen for
ive one fol
easy to
.= group velocity is
the farmers, and
OCR for page 455
egative for the tatters. So it follows
capillary
opposite
erections L1,2J, as it is well known
from experimental observation.
n
that gravity waves and
ripples propagate in
d
2.7 A simplified formulation for the
linear gravitycapillarity problem
Now we reconsider the linear
formulation described in 2.2. Let us
assume:
~ c
Y= ~ k!
(38) ~ c
where the superscripts G and C indicate
the terms connected respectively to the
gravity and capillary waves; in fact,
as it has been shown in the preceding
section, in the capillaritygravity
problem two wave sistems are present:
the capillary waye preceding the body,
and the gravity one following it.
So we can write:
~(7x+:X) tY+tY
U({X ~ AX ~ ~ ~ (7 F? ) p <~=0
km ~ = km iV9t = 0
~ ~ _ oo ~ ~ —o()
= km ~ = 0
x_ +0o
Since the
can apply the
effects, solving
(39) U )4 _ Sty
(40) Mix + ~ ~ ~ p ~xX = 0
problem is linear, we
superposition of the
two splitted problems:
(41) ~ 1 ~{ 1 = 0
X ~ _=
(42) ~ SAC _ SAC
(43) ~ My +~: _ T Arc _ o
(44) ear l~cl=0
X .~=
These formulations can be strongly
simplified assuming:
(45)
~   C by) t~
Tc= _ (Gil
that is, the free surface is
sinusoidal. Of course, hypothesis (45)
can be a good approximation only for
the far field: the solution available
with this assumption can so be
considered asymptotical. We have:
( 4 6 ) gym + (~+ T ( Ad) ) ~4 _
(47) Myth (~+ T LACE) c
and, considering (35), we get finally:
(48) 74 = _ Yx
tJk~4
( 49 ) i,t,`+ ~ ye = ~
C TIC
(50) ~ = x
vhe
(51) ~ sky =0
OX y
In order to solve problem (48),
(49) condition (41) must be considered,
as well as condition (44) for problem
(50), (51).
3. Numerical procedure and results
The numerical solution of the
mathematical models given in 2.3, 2.S
and 2.7 is reached by means of a simple
layer formulation:
461
OCR for page 455
(52, T(r, 1~[~+Q' ~ ~ 1 ~ ~
in which
r = jp_
r' = tp_~
r" = lp_
with p field point, c~ , quay ~ ,
q ~ ~ source points, being. ~' the
image of ~ with respect to the undi_
sturbed free surface.
The surfacers and a local portion
of S are discretized, by means of the
classical collocation method E53, with
segments tangent to the boundaries.
3.1 The nonlinear gravity problem
In the numerical procedure the
free surface S is followed, step by
step, updating its discretization and,
of course, the influence matrices.
The iterative scheme consists of
two cycles. An 'internal' one, in which
the nonlinear problem given by (2), (5)
and (12) is solved with an iterative
procedure; when the solution of this
system satisfies the required accuracy,
the free surface is updated in the
'external' cycle by means of (13). At
the first step, to initialize the
procedure, the potential flow and the
free surface configuration are computed
with Dawson method.
The computat
double iteration
summarized as in t
tonal steps, for the
scheme, can be
he following:
 solve the linear problem (2)
(I), (6), (a), (I) to initialize
the numerical procedure; `~'
 ext~em)na1 loop (index m): 7<
~ are known on each bounder
element i;
3  internal loop (index j) : solve
the nonlinear system
(j+4 >
I ~ = 0
At,
2(j) (+1) Ajar)
II ~ ~ {a ~  o
t~ Hi ~ Hi
in which the quadratic terms are
considered explicitly;, and the
condition II is imposed on the
previous appproximation (m) of
the free surface configuration
when ma All/ l^~ is
lower then a pref~.xe;] tollerance
go to step 4 ;
4  external loop (index m) : calcu_
ration of the new approximation
of S. with
III ~   ~ U l' )
when make ~~` /~^ l is less
then a required tolerance, the
numerical procedure is considered
to be concluded.
Near the downstream bou
if the damping zone suggeste
present, an oscillatory be
been observed, though for al
grid points convergence
reached; t
evident as
as it coul
notice the
more rapid_
for wave he
obvious sit
quantity,
applicatior
The s
been used
formulatior
the effect
surface; t
instead
conside~abl
of the met
involving ~
We read
in (12) or i
the four p
scheme urn
p
to
ndary, also
d in (53 is
havior has
l the other
has been
his phenomenon becomes more
the Froude number increases,
d be expected. Moreover we
t convergence is reached
1 v for wave rag i stance than
rather
average
norm i r~ 1
the
app
geo
1 o c
be
exp
be
462
ight; this behavior
ce the former is an
can be useful for r
ame nume I ical strategy has
also for the nonlinear
(24), (25), which includes
s of viscosity at the free
he use of such formulation
of (12), (13) extends
y the range of applicability
hod 15 , though the terms
iscosity are rather ~mall.
cork that the term ~~`
n (25) is discretized with
Dints finite differences
~  r posed by Dawson; the
ownstream damping zone, in which a two
Dints operator is used, is fixed equal
a quarter of the wave length.
To validate the present method,
numerical procedure has been
lied to a submerged hydrofoil, whose
metrical characteristics find
ation in the fluid dynamic field can
found in fig.1; several
erimental results for this case can
found in r33.
In all the examples the free
boundary has been discretized by 250
panels, and the boundary of the body by
40 elements. The number of panels per
wave length has been chosen equal to
40, and about 80 free surface elements
hare beeen L?lacecl upstream the boa:
leading edge. In the first two examples
we compute the wave pattern
corresponding to two different Froude
numbers (.4406 and .~`l7), with the
Froude number defined as If/ ~ (L is
the chord lent). The ;~ave pattern,
computed is the nonlinear p~ocedu~e
described before, by means of (is), is
compared with the experimental data
OCR for page 455
obtained in [3] and with the linear
numerical solution computed by Dawson's
method [5]. As it's shown in fig.2 and
3 the full nonlinear formulation gives
a numerical solution which fits the
experimental data better than the
linear one. We notice that in this
range of Froude number the inviscid
nonlinear formulation gives almost the
same results of (25), the only
difference being a small damping in the
wave amplitude for the viscous
formulation, as it can be expected.
This behavior is shown in fig.4
(Fr=.704), in which the typical steep
wave shape can be recognized; the
damping appears to be significant
rather far from the body: at a distance
about 10*L a 4 per cent attenuation is
present.
In fig.5 the wave resistance as a
function of the Prude number is plotted
(for this figure, Fr=U/Vg~h): also in
this case the nonlinear method behaves
better then the linear one, with
respect to the experimental data.
We remark that the inviscid
formulation gives practically the same
results for the wave resistance, but
its range of applicability is narrower
with respect to (25), in fact for
Fr>.57 the numerical procedure diverges
E ~ while the use of (25) allows one
to obtains results up to Fr=.65.
3.2 The linear gravitycapillarity
problem
We solve numerical!: both the
splitted problems (2), (5), t48), (49),
(41) and (2), (5), (50), (51), (44), by 3
means of D;~lvson r S method, noticing that
for the capillarit: problem the second
derivative must be discr*tized with .
downstream finite differences operator,
in order to satisfy the radiation
condition.
In fig.6 the wave pattern obtained
with this procedure is shown: the
dashed liners indicate the .splitte'l
solutions, while the continuous line
indicate the superposition of them. In
this case the body is a circular curling
der (diameter d = (?.01 m), with ~ depth
of the center equal to the diameter.
Surface tension value is .0/4 N/m; for
the discretization, 2~, panels per
capillary wale length hare been chosen,
with ~ total ntlml~~~ `,f I)) panels used.
the
to
with
Fi~.7 shows the behavior of
wave lengths accuracy with respect
the discretizatron, in com~ison
the analytical Values.
lineally, in fig.X the ~r;'lues
gravity anal capillary bale lengths
functions of the velocity I!
plotters, in comparison with
f
~ s
trot
the
dispersion relation (35); in this case,
30 panels per wave length have been
used.
Conclusions
Both the proposed numerical models
seem to be promising and suitable to
have more investigations. At the
present the introduction of viscosity
effects is rather effective in order to
get convergence with highly nonlinear
boundary conditions. The model with
surface tension can be a very deep tool
for describing nonlinear energy
transfer phenomena; the results
obtained with the simplified linear
model are preliminary steps for
studying a radiation condition suitable
also for the nonlinear model.
Acknowledgements
We wish to acknowledge Prof. P.
Bassanini, Dept. of Mathematics, Univ.
'La Sapienza' of Rome, for the helpful
suggestions and encouragements, and
Eng. T. Coppola, who implemented the
computational codes.
References

9
463
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~ Methods for
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OCR for page 455
Wave Resistance
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Radiati
State
of St
n.1., 1
1 Rong, H.
rical ~
ShipWa
?~/.
2 Sclavour
"Stabil
thods f
Forward
Naval H
Raven,
Theme ~
Naval H
14 Nakatake
J., Kat
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5 Campana,
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or FreeSurface ~
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arelli, U.,"Fully
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hangai, 1989.
y
, D. E.,
anel Me_
~lows with
rposium on
1988.
ns on a
IpOsium on
1988.
, Andou,
ation of
tiny on
, 1988.
Pitolli,
Nonlinear
ation by
method" ,
, f REE SURFACE
~R~ ~
~ .l4e .175 .te' .152 .1's .t48 .tt~
1 ~ ~
~ o .l .t ·' ·4 ~ ~ ·' ~ ~o
~1\ ~ ~
— 1
h=~.S
~ 1

Fig.l: Cross section of the hydrofoil [33;
quantities expressed in ft.
464
OCR for page 455

o 

~
cM
o 
o 
c~ 
o
o
· 
o
o
o"
IS on
 ~
I or r I
4. on S 00
lo
_
_
o—
O_ ~ I"
\ a/
\ ,~1
o
. _
o
~ At,
hair
, ~ A,
At,, ~ ah.
~ Am'
An'
q
I'll
1
/I
O I ~ ~ 1 1 1 1 1 1 1 1 1
O7S 1.3S 1.95 2.5S 3.15 3.75 4.3S
TIC
A< At
/';~'
"a ~
i'= .~' ~
_~
ARC ~
~ ARC
\',_.
X I L
~ ~ ~ 1 ~ _
4 95 5. SS 6.1 5
Fige 2: U/Vg*L S . 4406

Fir' <)
~ a'
~~ 5
.7
f>~
'\'
\\~
  ~ ~
~1
7
~r
Al
~c
~r
4
r I
6 04)
XtL
1 
7 00 6.Ot) 9 t)0 10.00 11 ·00 1200
Fig.3: U/V~ = .617
Waves generated by t!,e submerged hydrofoil, ~che arrows indicate the
posi~cion of the trailing cdge.
Dawson method
full nonlinear method
****~** experimental [33
465
OCR for page 455
o
 
~~ 
Q
_
o
C:
o
+_
1
U.
ID—
o
o
U.—
C'
o
~~ _
o
)
O
OCR for page 455
0
~ 
~o

 ~
· ~
~ 
o
o
  ~
to
o 
0~80 0.64 t) 48 O 32 0.46 O.OC 0.46 0~32
,,< 1t'1 ~ ~ (~N
. . ~ ~ 1
0.48 0.64 0.~0
Fig. 6: Gravitycapillarity steady free surface flow past a submerged
circular cyi inder:
 gravity and capillary waves
— superposition
o,—

_
:~ —
0 _
~ At_
~ _
~ m_
_ 1 ~

~ 00 ~ 6 00
r ~ I I I I r
32.00 &8 . OC 64
Fig. 7: Wave lengths versus N = number of panels
U = .234 m/s per wavelength;
L = .01 m
h = ,01 m
Fox
~ ru
,
c~ .
_ ~
c'
._ 1
me ~ ~ 1 O 1 (~)
Fig .8 ~ Dispe2~sion relation ( 35 )
L = .01 m

h = .01 m
467
analytical curve
oooooooo nwnerical values